Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(7,10)(8,12)(14,16)(17,18)(19,20)(21,29)(22,24)(23,25)(26,31)(27,30)(28,32)(33,34,38,42)(35,36,40,39)(37,45,47,41)(43,48,46,44);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19,22,25)(18,23,24,20)(21,27,31,32)(26,30,29,28)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,48)(46,47);(3,9)(7,10)(11,15)(14,16)(18,24)(20,23)(26,29)(28,30)(33,35,38,40)(34,39,42,36)(37,43,47,46)(41,48,45,44);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,20,22,23)(18,25,24,19)(21,30,31,28)(26,27,29,32)(33,35)(34,36)(37,46)(38,40)(39,42)(41,48)(43,47)(44,45);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,18)(19,20)(21,26)(22,24)(23,25)(27,28)(29,31)(30,32)(33,36)(34,40)(35,42)(37,44)(38,39)(41,46)(43,45)(47,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21,22,31)(18,26,24,29)(19,27,25,32)(20,28,23,30)(33,37,38,47)(34,41,42,45)(35,43,40,46)(36,44,39,48)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=i,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9575515);